Constructing and exploring wells of energy landscapes

Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features ``wells of duration $T$ and altitude $y$''. We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) which have been developed during the last two decades; we therefore propose a minimal appendix exposing them at the end of this paper to bridge the possible gap.Comment: 28 pages, submitted to J. Math. Phys -

Rescue, rehabilitation, and release of marine mammals: An analysis of current views and practices.

Stranded marine mammals have long attracted public attention. Those that wash up dead are, for all their value to science, seldom seen by the public as more than curiosities. Animals that are sick, injured, orphaned or abandoned ignite a different response. Generally, public sentiment supports any effort to rescue, treat and return them to sea. Institutions displaying marine mammals showed an early interest in live-stranded animals as a source of specimens -- in 1948, Marine Studios in St. Augustine, Florida, rescued a young short-finned pilot whale (Globicephala macrorhynchus), the first ever in captivity (Kritzler 1952). Eventually, the public as well as government agencies looked to these institutions for their recognized expertise in marine mammal care and medicine. More recently, facilities have been established for the sole purpose of rehabilitating marine mammals and preparing them for return to the wild. Four such institutions are the Marine Mammal Center (Sausalito, CA), the Research Institute for Nature Management (Pieterburen, The Netherlands), the RSPCA, Norfolk Wildlife Hospital (Norfolk, United Kingdom) and the Institute for Wildlife Biology of Christian-Albrects University (Kiel, Germany).(PDF contains 68 pages.

K to pi and K to 0 in 2+1 Flavor Partially Quenched Chiral Perturbation Theory

We calculate results for K to pi and K to 0 matrix elements to next-to-leading order in 2+1 flavor partially quenched chiral perturbation theory. Results are presented for both the Delta I=1/2 and 3/2 channels, for chiral operators corresponding to current-current, gluonic penguin, and electroweak penguin 4-quark operators. These formulas are useful for studying the chiral behavior of currently available 2+1 flavor lattice QCD results, from which the low energy constants of the chiral effective theory can be determined. The low energy constants of these matrix elements are necessary for an understanding of the Delta I=1/2 rule, and for calculations of epsilon'/epsilon using current lattice QCD simulations.Comment: 43 pages, 2 figures, uses RevTeX, added and updated reference

Quantum Pumping with Ultracold Atoms on Microchips: Fermions versus Bosons

We present a design for simulating quantum pumping of electrons in a mesoscopic circuit with ultra-cold atoms in a micro-magnetic chip trap. We calculate theoretical results for quantum pumping of both bosons and fermions, identifying differences and common features, including geometric behavior and resonance transmission. We analyze the feasibility of experiments with bosonic $^{87}$Rb and fermionic $^{40}$K atoms with an emphasis on reliable atomic current measurements.Comment: 4 pages; 4 figure

Study of Solutions to Differential Inclusions by the "Pipe Method"

A pipe of a differential inclusion is a set-valued map associating with each time t a subset P(t) of states which contains a trajectory of the differential inclusion for any initial state x_o belonging to P(O). As in the Liapunov method, knowledge of a pipe provides information on the behavior of the trajectory. In this paper, the characterization of pipes and non-smooth analysis of set-valued maps are used to describe several classes of pipes. This research was conducted within the framework of the Dynamics of Macrosystems study in the System and Decision Sciences Program

Contingent Isaacs Equations of a Differential Game

The purpose of this paper is to characterize classical and lower semicontinuous solutions to the Hamilton-Jacobi-Isaacs partial differential equations associated with a differential game and, in particular, characterize closed subsets the indicators of which are solutions to these equations. For doing so, the classical concept of derivative is replaced by contingent epi-derivative, which can apply to any function. The use of indicator of subsets which are solutions of either one of the contingent Isaacs equation allows to characterize areas of the playability set in which some behavior (playability, winability, etc.) of the players can be achieved

Qualitative Differential Games: A Viability Approach

The author defines the playability property of a qualitative differential game, defined by a system of differential equations controlled by two controls. The rules of the game are defined by constraints on the states of each player depending on the state of the other player. This paper characterizes the playability property by a regulation map which associates with any playable state a set of playable controls. In other words, the players can implement playable solutions to the differential game by playing for each state a static game on the controls of the regulation subset. One must extract among theses playable controls the set of discriminating and pure controls of one of the players. Such controls are defined through an adequate "contingent" Hamilton-Jacobi-Isaacs equation. Sufficient conditions implying the existence of continuous or minimal playable, discriminating and pure feedbacks are provided

Smooth and Heavy Solutions to Control Problems

We introduce the concept of viability domain of a set-valued map, which we study and use for providing the existence of smooth solutions to differential inclusions. We then define and study the concept of heavy viable trajectories of a controlled system with feedbacks. Viable trajectories are trajectories satisfying at each instant given constraints on the state. The controls regulating viable trajectories evolve according a set-valued feedback map. Heavy viable trajectories are the ones which are associated to the controls in the feedback map whose velocity has at each instant the minimal norm. We construct the differential equation governing the evolution of the controls associated to heavy viable trajectories and we state their existence

Slow and Heavy Viable Trajectories of Controlled Problems. Part 1. Smooth Viability Domains

We define slow and heavy viable trajectories of differential inclusions and controlled problems. Slow trajectories minimize at each time the norm of the velocity of the state (or the control) and heavy trajectories the norm of the acceleration of the state (or the velocity of the control). Macrosystems arising in social and economic sciences or biological sciences seem to exhibit heavy trajectories. We make explicit the differential equations providing slow and heavy trajectories when the viability domain is smooth